Simplify and expand the following expression: $ \dfrac{5}{p + 2}+\dfrac{p + 4}{3p + 3} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(p + 2)(3p + 3)$ Multiply the first term by $\dfrac{3p + 3}{3p + 3}$ $ \begin{align*} \dfrac{5}{p + 2} \times \dfrac{3p + 3}{3p + 3} & = \dfrac{(5)(3p + 3)}{(p + 2)(3p + 3)} \\ & = \dfrac{15p + 15}{(p + 2)(3p + 3)}\end{align*} $ Multiply the second term by $\dfrac{p + 2}{p + 2}$ $ \begin{align*} \dfrac{p + 4}{3p + 3} \times \dfrac{p + 2}{p + 2} & = \dfrac{(p + 4)(p + 2)}{(3p + 3)(p + 2)} \\ & = \dfrac{p^2 + 6p + 8}{(3p + 3)(p + 2)}\end{align*} $ Now we have: $ = \dfrac{15p + 15}{(p + 2)(3p + 3)} + \dfrac{p^2 + 6p + 8}{(3p + 3)(p + 2)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{15p + 15 + p^2 + 6p + 8}{(p + 2)(3p + 3)} $ $ = \dfrac{21p + 23 + p^2}{(p + 2)(3p + 3)}$ Expand the denominator: $ = \dfrac{21p + 23 + p^2}{3p^2 + 9p + 6}$